Feedback Stabilization

In principle, all the above-mentioned noise sources double as potential control pa-rameters for a feedback stabilization scheme, as they all affect the intracavity dis-persion. The most straightforward implementation of dispersion control is realized simply by changing the amount of dispersive material in the optical path, which is typically done with a pair of moveable glass wedges [105]. While this method allows for large phase shifts of several cycles, the inertia of the optical components strongly limits the control bandwidth, which applies to all stabilization schemes with mechanically moving parts. An exception to this rule is a method that intro-duces geometrical dispersion by tilting a cavity mirror in a prism-based oscillator.

This approach was shown to support control bandwidths of up to 25 kHz [106] but is only applicable to this special type of oscillators that is not widely used anymore.

Certainly the most common feedback mechanism relies on the modulation of the oscillator pump power, which affects the CEP by the complex interplay of various processes. Apart from the thermal influence on the GPO [26], the CEP is changed by a combination of a power-dependent spectral shift and a non-vanishing GDD [11]. Another contribution to the GPO stems from the dispersion of the nonlinear phase shift that is predominantly acquired in the laser crystal [26, 90]. While the interplay of these processes and therefore the observed overall power dependence

2.5 CEP Stabilization

strongly depends on the specific oscillator design and the exact conditions, it has been shown that the nonlinear dispersion effectn₂(ω) is the dominant contribution in modern dispersion-managed Ti:sapphire oscillators [107].

The key advantage of the nonlinear feedback mechanism is the intrinsically fast response time, which is, in principle, only limited by the laser dynamics that con-vert the pump power changes into intracavity peak power fluctuations. In practice, however, the achievable control bandwidth is mostly constrained by the servo elec-tronics and the typically employed acousto-optical power modulation, which still enable control bandwidths on the order of 100 kHz [108]. The downside of this feedback mechanism is that the action on the CEP is accompanied by an ampli-tude modulation of the oscillator pulse train. Given that the B integral of most Ti:sapphire oscillators is on the order of 2π, the pump power induced phase shift is limited to some 100 mrad before the disturbance of the oscillator becomes too severe to sustain the mode-locked operation.

In fact, it is a characteristic feature of all feedback mechanisms, that side-effects on other laser parameters, such as repetition rate, output power or beam point-ing, prevent a CEP stabilization without interrupting the performance of the free-running oscillator. In the next section 2.5.2, an extra-cavity feed-forward stabiliza-tion scheme will be presented that overcomes most of these limitastabiliza-tions.

In order to ensure CEP stabilization via the feedback approach, first a suitable error signal has to be deduced from the measured in-loop beat signal. As the measurement and the stabilization of CEP noise at the DC baseband is prevented in RF-heterodyning schemes, the interferometrically detected beat signal Vf−2f is typically locked to another RF reference oscillator. To this end, the beat signal at f_CE is filtered and compared to the reference signal V_ref with a phase detector. In case of analog signal processing the phase comparison of two signals is accomplished by an electronic multiplication in a double-balanced mixer and subsequent low-pass filtering, see Fig. 2.7. Assuming the beat signal and the reference signal are given by pure sinusoidal oscillations

V_f−2f = sin(ω_CEt+ϕ_CE+ϕ₀) (2.5.1) V_ref = sin(ω_reft+ϕ_ref), (2.5.2) the multiplied signal will contain mixing products at ω_CE+ω_ref and ω_CE−ω_ref. After low-pass filtering an error signal is yielded that reads

V_err∝sin (ω_CEt+ϕ_CE+ϕ₀−ω_reft−ϕ_ref). (2.5.3) If the beat frequency approaches the reference frequency (ωCE → ωref), the error signal is reduced to a voltage that is proportional to the phase difference of the beat signal and the reference.

V_err∝sin(∆ϕ)≈∆ϕ for ∆ϕπ. (2.5.4)

In the simplest case the control loop is closed by using this error signal directly to change the feedback parameter. This way, the phase difference is diminished until the error signal eventually shows just some residual jitter, which is referred to as a phase-locked loop (PLL), see Fig. 2.7. In practice, a PLL is equipped with an additional servo controller that further processes the error signal with an adjustable proportional, integral and differential element, in order to improve the stability and the performance of the lock. From the perspective of control theory, a stability criterion for a PLL is a transfer function with a positive phase margin, meaning that the phase difference between input and output signal is less than π at the frequency of unity loop gain [108]. Otherwise, phase errors experience amplification instead of damping and self-oscillations occur. A limiting factor of such analog PLLs is the occurrence of phase ambiguities if the phase errors exceed the interval [−π/2, π/2]. These ambiguities can produce cycle slips, or in the worst case force the PLL to unlock. To this end, either frequency division of the beat signal prior to the analog phase detection, or completely digital phase detection is often used to increase the phase ambiguity range [109]. While such measures render a PLL more robust against drop-outs, there is an intrinsic trade-off between the tolerance towards large phase errors and the achievable phase resolution that influences the residual phase jitter.

For many applications, e.g., for seeding a subsequent amplification stage, it is desirable to use a reference oscillator that is related to the laser repetition rate. A reference at the full repetition rate and integer multiples thereof creates the same issues as the stabilization at DC, namely the inability to distinguish the beat signal from the always present intermode beat and the ambiguity of positive and negative frequency excursions. Subharmonics of the repetition rate (f_rep/N) on the other hand are readily filtered and used as a reference, thereby creating pulse trains in which everyNth pulse yields the same field structure.

Ti:Sa

Figure 2.7: Phase-locked-loop (PLL) for locking the CEP drift to a reference os-cillator. The phase error signal is derived by electronic mixing and subsequent lowpass filtering. This signal is further processed with a servo before it is fed back to the laser oscillator. Optionally a subhar-monic of the repetition rate may be used as a reference signal (dashed lines).

2.5 CEP Stabilization